Finite Group Databasebeta

← #161108

Subgroups — #161108

Order 512.

978 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34330C211yesmaximal, characteristic
2#22839C211yesmaximal, characteristic, complemented
3#22008C211yesmaximal, characteristic, complemented
4#7647C212yesmaximal, complemented
5#7645C212yesmaximal, complemented
6#7620C212yesmaximal, complemented
7#7612C212yesmaximal, complemented
8#7587C211yesmaximal, characteristic, complemented
9#7577C211yesmaximal, characteristic, complemented
10#7569C211yesmaximal, characteristic, complemented
11#7562C211yesmaximal, characteristic, complemented
12C2×#550C411yescharacteristic
13#2366—21no—
14#2366C411yescharacteristic
15#2364—22no—
16#2363C412yes—
17#2357—21no—
18#2355C411yescharacteristic
19#2354C411yescharacteristic
20#2339C412yes—
21#2333—22no—
22#2328—21no—
23C4×#127C2211yescharacteristic
24C4×#121C2212yes—
25#2297C2212yes—
26#2295C2211yescharacteristic
27#2275—21no—
28C42×D4C2211yescharacteristic
29#1954—21no—
30#1954—21no—
31#1939—21no—
32#1924—22no—
33#1923—22no—
34#1831C2211yescharacteristic
35#1817C2212yes—
36#1817C2212yes—
37#1797C2212yes—
38#1770—21no—
39C4×C4⋊C8C2211yescharacteristic
40C4×C4⋊C8C2211yescharacteristic
41#1749C2212yes—
42#1744C2211yescharacteristic
43#1272C2211yescharacteristic
44#1272C2211yescharacteristic
45#1272C2212yes—
46#1271C2211yescharacteristic
47#1271C2211yescharacteristic
48#1271C2212yes—
49#1270C2212yes—
50#1270C2212yes—
51#1270C2212yes—
52#1270C2212yes—
53C22×#41C2×C412yes—
54#550—24no—
55#550—24no—
56C2×#127—21no—
57C2×#127C2×C411yescharacteristic
58C2×#121—22no—
59C2×#121C2×C412yes—
60C2×C4×Q8C2×C411yescharacteristic
61C2×C4×D4—22no—
62C2×C4×D4—21no—
63C2×C4×D4—21no—
64C2×C4×D4—21no—
65C2×C4×D4—21no—
66C2×C4×D4C2×C412yes—
67C2×C4×D4C2×C412yes—
68C2×C4×D4C2×C411yescharacteristic
69C2×C4×D4D411yescharacteristic
70C2×C4×D4D411yescharacteristic
71C2×#117C2×C412yes—
72C22×C42C2312yes—
73C2×C4⋊C8—21no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8—21no—
76C2×C4⋊C8—22no—
77C2×C22⋊C8—22no—
78C2×C22⋊C8—22no—
79C2×C22⋊C8—22no—
80C2×C22⋊C8—22no—
81C2×#97—22no—
82C2×C4×C8—21no—
83C2×C4×C8—21no—
84C2×C4×C8—21no—
85#391C2×C411yescharacteristic
86#391C2×C412yes—
87#391C2311yescharacteristic
88#390—21no—
89#390—21no—
90#390—22no—
91#390—21no—
92#390—21no—
93#390C2×C412yes—
94#390C2312yes—
95#389C2×C411yescharacteristic
96#389C2311yescharacteristic
97#388—22no—
98#388—22no—
99#388—22no—
100#387—21no—
101#387—21no—
102#387—22no—
103#386—22no—
104#386—22no—
105#386—22no—
106#386—22no—
107#386—22no—
108#386—22no—
109#386—22no—
110#384C2×C411yescharacteristic
111#382—21no—
112#381—22no—
113#380C2×C412yes—
114C4×C4⋊C4—21no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4C2×C411yescharacteristic
118C4×C4⋊C4C2×C412yes—
119C4×C4⋊C4C2×C411yescharacteristic
120C4×C4⋊C4D411yescharacteristic
121C4×C4⋊C4D411yescharacteristic
122C4×C22⋊C4—22no—
123C4×C22⋊C4—22no—
124C4×C22⋊C4—22no—
125C4×C22⋊C4—22no—
126C4×C22⋊C4C2×C412yes—
127C4×C22⋊C4D412yes—
128C4×C22⋊C4D412yes—
129#376—21no—
130C2×#95—22no—
131C43C2311yescharacteristic
132#336—22no—
133#336—22no—
134#336—21no—
135#336—21no—
136#336C2311yescharacteristic
137#336C2312yes—
138#336C2311yescharacteristic
139#336C2311yescharacteristic
140#336C2312yes—
141#336C2311yescharacteristic
142C2×#41—24no—
143C2×#41—24no—
144C2×#41—82no—
145C2×#41—82no—
146C22×Q8C22×C411yescharacteristic
147C22×D4—22no—
148C22×D4—22no—
149C22×D4—21no—
150C22×D4—21no—
151C22×D4C22⋊C411yescharacteristic
152C22×D4C22×C412yes—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4C22×C412yes—
157C23×C4C2×D412yes—
158C23×C4C2×D412yes—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—22no—
163C22×C8—21no—
164C22×C8—22no—
165#127—28no—
166#127—22no—
167#127M1614yes—
168#121—216no—
169#121—24no—
170#121—24no—
171C4×Q8—22no—
172C4×Q8M1614yes—
173C4×D4—28no—
174C4×D4—22no—
175C4×D4—22no—
176C4×D4—22no—
177C4×D4—22no—
178C4×D4—42no—
179C4×D4—44no—
180C4×D4—42no—
181C4×D4—42no—
182C4×D4—81no—
183C4×D4—81no—
184C4×D4—82no—
185C4×D4—82no—
186C4×D4—82no—
187C4×D4—82no—
188#117—82no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4C22⋊C411yescharacteristic
195C2×C4⋊C4C22⋊C411yescharacteristic
196C2×C4⋊C4C22⋊C411yescharacteristic
197C2×C4⋊C4C22×C411yescharacteristic
198C2×C4⋊C4C22×C411yescharacteristic
199C2×C4⋊C4C22×C412yes—
200C2×C4⋊C4C22×C411yescharacteristic
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4C22⋊C412yes—
211C2×C22⋊C4C22⋊C412yes—
212C2×C22⋊C4C22⋊C412yes—
213C2×C42—21no—
214C2×C42—21no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—82no—
218C2×C42C22×C411yescharacteristic
219C2×C42C22×C412yes—
220C2×C42C22×C411yescharacteristic
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D411yescharacteristic
223C2×C42C2×D411yescharacteristic
224C2×C42C2×D412yes—
225C2×C42C2×D412yes—
226C2×C42C2×D411yescharacteristic
227C2×C42C2411yescharacteristic
228C4⋊C8—42no—
229C4⋊C8—42no—
230C4⋊C8—42no—
231C4⋊C8—82no—
232C22⋊C8—44no—
233C22⋊C8—82no—
234C22⋊C8—82no—
235C22⋊C8—82no—
236#97—82no—
237C4×C8—42no—
238C4×C8—42no—
239C4×C8—81no—
240#95—21no—
241#95—22no—
242#95—22no—
243#95—21no—
244#95—22no—
245#95—21no—
246#95—22no—
247#95—21no—
248#95—21no—
249#95—21no—
250#95—21no—
251#95C22⋊C412yes—
252#95C22⋊C412yes—
253#95C22⋊C412yes—
254C24—22no—
255C24C2×C22⋊C412yes—
256#41—88no—
257#41—88no—
258C2×Q8—21no—
259C2×Q8—41no—
260C2×Q8—41no—
261C2×Q8C2×M1612yes—
262C2×D4—28no—
263C2×D4—28no—
264C2×D4—22no—
265C2×D4—22no—
266C2×D4—24no—
267C2×D4—24no—
268C2×D4—22no—
269C2×D4—22no—
270C2×D4—42no—
271C2×D4—42no—
272C2×D4—42no—
273C2×D4—42no—
274C2×D4—81no—
275C2×D4—81no—
276C2×D4—82no—
277C2×D4—82no—
278C2×D4—82no—
279C2×D4—82no—
280C2×D4C2×M1614yes—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—28no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—22no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—44no—
300C22×C4—44no—
301C22×C4—82no—
302C22×C4—82no—
303C22×C4—82no—
304C22×C4—82no—
305C22×C4—82no—
306C22×C4—82no—
307C22×C4C2×C22⋊C412yes—
308C22×C4C2×C22⋊C412yes—
309C22×C4C2×C22⋊C412yes—
310C22×C4C2×C22⋊C411yescharacteristic
311C22×C4C2×C22⋊C411yescharacteristic
312C22×C4C2×C22⋊C411yescharacteristic
313C22×C4C2×C22⋊C411yescharacteristic
314C22×C4C23×C411yescharacteristic
315C22×C4C22×D411yescharacteristic
316C22×C4C22×D411yescharacteristic
317C2×C8—22no—
318C2×C8—41no—
319C2×C8—41no—
320C2×C8—41no—
321C2×C8—41no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8—42no—
327C2×C8—42no—
328C2×C8—41no—
329C2×C8—41no—
330C2×C8—41no—
331C2×C8—42no—
332C2×C8—42no—
333C2×C8—41no—
334C2×C8—41no—
335C4⋊C4—21no—
336C4⋊C4—24no—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—41no—
341C4⋊C4—42no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C4⋊C4—41no—
345C4⋊C4—42no—
346C4⋊C4—81no—
347C4⋊C4—81no—
348C4⋊C4C2×M1612yes—
349C22⋊C4—28no—
350C22⋊C4—24no—
351C22⋊C4—24no—
352C22⋊C4—42no—
353C22⋊C4—44no—
354C22⋊C4—42no—
355C22⋊C4—42no—
356C22⋊C4—42no—
357C22⋊C4—42no—
358C22⋊C4—82no—
359C22⋊C4—82no—
360C22⋊C4—82no—
361C22⋊C4—82no—
362C22⋊C4—82no—
363C42—21no—
364C42—21no—
365C42—24no—
366C42—22no—
367C42—42no—
368C42—41no—
369C42—42no—
370C42—41no—
371C42—42no—
372C42—41no—
373C42—42no—
374C42—82no—
375C42—82no—
376C42C2×M1612yes—
377C42C2×M1612yes—
378C42#13512yes—
379C42#13512yes—
380C23—22no—
381C23—24no—
382C23—22no—
383C23—22no—
384C23—22no—
385C23—22no—
386C23—22no—
387C23—22no—
388C23—42no—
389C23—42no—
390C23—42no—
391C23—42no—
392C23—42no—
393C23—42no—
394C23C22×C22⋊C411yescharacteristic
395Q8—42no—
396Q8—42no—
397D4—44no—
398D4—44no—
399D4—44no—
400D4—48no—
401D4—84no—
402D4—88no—
403D4—88no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—21no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—21no—
425C2×C4—22no—
426C2×C4—21no—
427C2×C4—22no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—88no—
448C2×C4—164no—
449C2×C4—164no—
450C2×C4#40712yes—
451C2×C4#40712yes—
452C2×C4#41612yes—
453C2×C4#41612yes—
454C2×C4#42112yes—
455C2×C4C22×M1611yescharacteristic
456C2×C4C22×M1611yescharacteristic
457C2×C4C2×#13511yescharacteristic
458C2×C4C2×#13511yescharacteristic
459C2×C4#57611yescharacteristic
460C2×C4#57711yescharacteristic
461C8—42no—
462C8—81no—
463C8—82no—
464C8—81no—
465C8—81no—
466C8—82no—
467C22—42no—
468C22—42no—
469C22—44no—
470C22—42no—
471C22—44no—
472C22—42no—
473C22—42no—
474C22—42no—
475C22—42no—
476C22—44no—
477C22—82no—
478C22—82no—
479C22—82no—
480C22—82no—
481C22C2×#40711yescharacteristic
482C22#288311yescharacteristic
483C22#288811yescharacteristic
484C22C2×#41611yescharacteristic
485C22#289711yescharacteristic
486C22C2×#42111yescharacteristic
487C22#290111yescharacteristic
488C4—22no—
489C4—22no—
490C4—21no—
491C4—21no—
492C4—21no—
493C4—21no—
494C4—21no—
495C4—21no—
496C4—41no—
497C4—42no—
498C4—42no—
499C4—42no—
500C4—42no—
501C4—42no—
502C4—81no—
503C4—82no—
504C4—81no—
505C4#148814yes—
506C2—82no—
507C2—84no—
508C2#1736911yescharacteristic
509C2C2×#148811yescharacteristic
510C2#1745311yescharacteristic
511C2#1759911yescharacteristic
512C2#1761711yescharacteristic
513C2#1768411yescharacteristic
514C2#1772411yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#550C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #22839C2semidirect
3. #22008C2semidirect
4. #7647C2semidirect
5. #7645C2semidirect
6. #7620C2semidirect
7. #7612C2semidirect
8. #7587C2semidirect
9. #7577C2semidirect
10. #7569C2semidirect
11. #7562C2semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#343302
#228392
#220082
#76472
#76452
#76202
#76122
#75872
#75772
#75692
#75622